CIIM 2012

For each positive integer $n$ let $A_n$ be the $n \times n$ matrix such that its $a_{ij}$ entry is equal to ${i+j-2 \choose j-1}$ for all $1\leq i,j \leq n.$ Find the determinant of $A_n$ .

A set $A\subset \mathbb{Z}$ is "padre" if whenever $x,y \in A$ with $x\leq y$ then also $2y -x \in A$ . Prove that if $A$ is "padre", $0,a,b \in A$ with $0< a < b$ and $d = g.c.d(a,b)$ then $$a+b-3d, a+b-2d \in A.$$

Let $a,b,c,$ the lengths of the sides of a triangle. Prove that $$\sqrt{\frac{(3a+b)(3b+a)}{(2a+c)(2b+c)}} + \sqrt{\frac{(3b+c)(3c+b)}{(2b+a)(2c+a)}} + \sqrt{\frac{(3c+a)(3a+c)}{(2c+b)(2a+b)}} \geq 4.$$

Let $f(x) = \frac{\sin(x)}{x}$ Find $$\lim_{T\to\infty}\frac{1}{T}\int_0^T\sqrt{1+f'(x)^2}dx.$$

Let $D=\{0,1,\dots,9\}$ . A direction function for $D$ is a function $f:D \times D \to \{0,1\}.$ A real $r\in [0,1]$ is compatible with $f$ if it can be written in the form $$r = \sum_{j=1}^{\infty} \frac{d_j}{10^j}$$ with $d_j \in D$ and $f(d_j,d_{j+1})=1$ for every positive integer $j$ .

Determine the least integer $k$ such that for any direction fuction $f$ , if there are $k$ compatible reals with $f$ then there are infinite reals compatible with $f$ .

Let $n \geq 2$ and $p(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$ a polynomial with real coefficients. Show that if there exists a positive integer $k$ such that $(x-1)^{k+1}$ divides $p(x)$ then $$\sum_{j=0}^{n-1}|a_j| > 1 +\frac{2k^2}{n}.$$